Implémentation et applications d'algorithmes fondés sur la théorie de la fonctionnelle de la densité dépendante du temps dans les logiciels à la base des fonctions gaussiennes et ondelettes

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Implémentation et applications d’algorithmes fondés sur la théorie de la fonctionnelle de la densité dépendante

du temps dans les logiciels à la base des fonctions gaussiennes et ondelettes

Bhaarathi Natarajan

To cite this version:

Bhaarathi Natarajan. Implémentation et applications d’algorithmes fondés sur la théorie de la fonc- tionnelle de la densité dépendante du temps dans les logiciels à la base des fonctions gaussiennes et ondelettes. Sciences agricoles. Université de Grenoble, 2012. Français. �NNT : 2012GRENV002�.

�tel-00682011�

Pour obtenir le grade de

DOCTEUR DE L’UNIVERSITÉ DE GRENOBLE

Spécialité : Chimie physique

Arrêté ministérial : 7 Août 2006

Présentée par

Bhaarathi Natarajan

Thèse dirigée par Mark Earl Casida et codirigée par Thierry Deutsch

préparée au sein Département de Chimie Moléculaire (DCM) et de Chimie et Science du Vivant (CSV)

Implementation, Testing, and Appli- cation of Time-Dependent Density- Functional Theory Algorithms for Gaussian- and Wavelet-based Pro- grams

Thèse soutenue publiquement le Laboratoire de Chimie Théorique, devant le jury composé de :

Prof. Éric Suraud

Laboratoire de Physique Théorique - IRSAMC University Paul Sabatier Toulouse 3 118 Route de Narbonne, 31062 Toulouse cedex , Président

Prof. Dietrich Förster

CPMOH, University of Bordeaux 1, 351 Cours de la Liberation, 33405 Talence, France, Rapporteur

Prof. Jürg Hutter

Physical Chemistry Institute, University of Zurich, Winterthurerstrasse 190, CH- 8057 Zurich, Switzerland, Rapporteur

Dr. Valérie Véniard

Laboratoire des Solides Irradiés, École Polytechnique, 91128 Palaiseau Cedex, France, Examinateur

Dr. Miguel Alexandre Lopes Marques

Laboratoire de Physique de la Matière Condensée et Nanostructures, Univer- sité Lyon 1 et CNRS, 43 blv. du 11 novembre 1918, 69622 Villeurbanne Cedex, France, Examinateur

Mark E. Casida

Laboratoire de Chimie Théorique, Département de Chimie Moléculaire (DCM), Université Joseph Fourier, Grenoble, France , Directeur de thèse

Thierry Deutsch

Institut Nanosciences et Cryogénie, SP2M/L-Sim, CEA cedex 9, 38054 Grenoble, FRANCE , Co-Directeur de thèse

Bhaarathi NATARAJAN: PhD Thesis, Implementation, Testing, and Ap-

plication of Time-Dependent Density-Functional Theory Algorithms

for Gaussian- and Wavelet-Based Programs , © March 3, 2012

The interaction of light with matter is a well-established domain of physical science. For a chemical physicist, this interaction may be used as a probe (spectroscopy) or to induce chemical reactions (photo- chemistry.) Photochemical reaction mechanisms are difficult to study experimentally and even the most sophisticated modern femtosecond spectroscopic studies can benefit enormously from the light of theoret- ical simulations. Spectroscopic assignments often also require theoret- ical calculations. Theoretical methods for describing photoprocesses have been developed based upon wave-function theory and show remarkable success when going to sophisticated higher-order approxi- mations. However such approaches are typically limited to small or at best medium-sized molecules. Fortunately time-dependent density- functional theory (TD-DFT) has emerged as a computationally-simpler method which can be applied to larger molecules with an accuracy which is often, but not always, similar to high-quality wave-function calculations. Part of this thesis concerns overcoming difficulties in- volving the approximate functionals used in present-day TD-DFT. In particular, we have examined the quality of conical intersections when the Ziegler-Wang noncollinear spin-flip approach is used and have shown that the spin-flip approach has merit as a particular solution in particular cases but is not a general solution to improving the de- scription of conical intersections in photochemical simulations based upon TD-DFT. Most of this thesis concerns algorithmic improvements aimed at either improving the analysis of TD-DFT results or extending practical TD-DFT calculations to larger molecules. The implementa- tion of automatic molecular orbital symmetry analysis in deMon2k is one contribution to improving the analysis of TD-DFT results. It also served as an introduction to a major programming project. The major methodological contribution in this thesis is the implementation of Casida’s equations in the wavelet-based code BigDFT and the subse- quent analysis of the pros and cons of wavelet-based TD-DFT where it is shown that accurate molecular orbitals are more easily obtained in BigDFT than with deMon2k but that handling the contribution of unoccupied orbitals in wavelet-based TD-DFT is potentially more problematic than it is in a gaussian-based TD-DFT code such as de- Mon2k. Finally the basic equations for TD-DFT excited state gradients are derived. The thesis concludes with some perspectives about future work.

v

R É S U M É

L’interaction entre la matiére et le rayonnement est un domaine bien établi de la physique. Pour un physico-chimiste, cette interaction peut être utilisée comme une sonde (spectroscopie) ou pour provoquer des réactions chimiques (photo-chimie). Les mécanismes des réac- tions photochimiques sont difficiles à étudier expèrimentalement et même les études les plus sophistiquées de spectroscopies femtosec- ondes peuvent bénéficier énormément des simulations théoriques.Les résultats spectroscopiques d’ailleurs ont souvent besoin des calculs théoriques pour l’analyse de leurs spectres. Les méthodes théoriques pour décrire les processus photochimiques ont été principalement développées en utilisant le concept de la fonction d’onde à N corps et ont eu des succès remarquables. Cependant de telles approches sont généralement limitées à des petites ou moyennes molécules. Heureuse- ment la théorie de la fonctionnelle de la densité dépendant du temps (TD-DFT) a émergé comme une méthode simple de calcul pouvant être appliquée à des molécules plus grandes, avec une précision qui est souvent, mais pas toujours, semblable à la précision provenant des méthodes basés sur la fonction d’onde à N électrons. Une partie de cette thèse consiste à surmonter les difficultés des approxima- tions utilisées de nos jours en TD-DFT. En particulier, nous avons examiné la qualité des intersections coniques quand l’approche du retournement de spin non collinéaire de Ziegler-Wang est utilisée et nous avons montré que l’approche du retournement de spin, parfois ,améliore dans des cas particuliers, mais que c’est n’est pas une so- lution générale pour mieux décrire les intersections coniques dans les simulations photochimiques basées sur la TD-DFT. La plupart des parties de cette thèse traite d’améliorations algorithmiques, soit pour améliorer l’analyse des résultats de la TD-DFT, soit pour étendre les calculs de TD-DFT à de grandes molécules. L’implémentation de l’analyse automatique des symétries des orbitales moléculaires dans deMon2k est une contribution pour améliorer l’analyse des résultats de la TD-DFT. Cela a aussi servi comme une introduction au projet de programmation majeur. La contribution méthodologique principale dans cette thèse est l’implémentation des équations de Casida dans le code BigDFT fondé sur le formalisme des ondelettes. Cette implé- mentation a aussi permis une analyse détaillée des arguments positifs et négatifs de l’utilisation de la TD-DFT fondée sur les ondelettes.

On montre qu’il est plus facile d’obtenir des orbitales moléculaires précises qu’avec deMon2k. Par contre, la contribution des orbitales inoccupées est plus problématiques qu’avec un code de gaussienne comme deMon2k. Finalement, les équations de base des gradients

vi

vii

1. Miquel Huix-Rotllant, Bhaarathi Natarajan, Andrei Ipatov, C.

Muhavini Wawire, Thierry Deutsch, and Mark E. Casida, Phys.

Chem. Chem. Phys., 12, 12811-12825 (2010). "Assessment of Noncollinear Spin-Flip Tamm-Dancoff Approximation Time- Dependent Density-Functional Theory for the Photochemical Ring-Opening of Oxirane"

2. Mark E. Casida, Bhaarathi Natarajan, and Thierry Deutsch, in Fundamentals of Time-Dependent Density-Functional Theory, edited by Miquel A. L. Marques, N. T. Maitra, Fernando Noguiera, E. K. U. Gross, and Angel Rubio (Springer: in press). "Non- Born-Oppenheimer Dynamics and Conical Intersections",

//arxiv.org/abs/1102.1849

3. Bhaarathi Natarajan, Luigi Genovese, Mark E. Casida, Thierry Deutsch, Olga N. Burchak, Christian Philouze, and Maxim Y.

Balakirev, Chem. Phys., under review "Wavelet-Based Linear- Response Time-Dependent Density-Functional Theory",

//arXiv:1108.3475v1

4. Bhaarathi Natarajan, Mark E. Casida, Luigi Genovese, and Thierry Deutsch, in Theoretical and Computational Developments in Modern Density Functional Theory, edited by A. K. Roy ( in press).

"Wavelets for Density-Functional Theory and for Post Depen- dent Density-Functional Theory Calculations",

1110.4853v1

ix

A C K N O W L E D G M E N T S

The work in this thesis is very much an overlapping project between two different departments (Physics and Chemistry) in two different institutes (CEA and University of Grenoble.) Consequently, there are many people I would like to thank, without whom I would never have reached this level of academic achievement. First among them are my supervisors Mark E. Casida and Thierry Deutsch, both of whom are extremently patient, knowledgeable and hard-working, yet always approachable. Especially I am very grateful to Mark for his expert advice and all his time and support. Mark’s wide knowledge and his logical way of thinking have been of great value for me.

His perpetual energy and enthusiasm in research have motivated all his students, including me. In addition, Mark is always accessible and willing to help his students with their research with his deep personal understanding. As a result, research life became smooth and rewarding for me. A special thanks to Dr. Luigi Genovese for fruitful discussions and for showing me the minute details of the BigDFT code.

During this work I have discussed with many experts for whom I have great regard, and I wish to extend my warmest thanks to all those who have helped me with my work, especially Andreas Köster, Souvrav Pal, Claude Paul, Klaus Hermann, and Damian Castile

Many other people have been helpful with practical advice and assistance during different periods of my PhD studies, among which I am infinitely grateful to Kim Casida, for her love and affection, Arpan Krishna Deb for very welcome coffee breaks, Radhika Ramadoss, and Anusha Muthukumar.

In addition, I would like to express my gratitude to the Nanoscience Foundation, and CEA for their financial and admistrative support over the last three years, without which I would never have been able to undertake this research project.

Finally, I am eternally indebted to my friends and family. Most importantly are my parents, their encouragement, understanding and guidance has been steadfast and I will be forever grateful for their support. I am also thankful to my close friends for keeping me grounded and helping me to unwind, especially over this period of thesis studies.

xi

List of Figures xvii List of Tables xxii

i theoretical motivation 1 1 background material 3

1.1 Synopsis 3 1.2 Atomic Units 5

1.2.1 Naming Excited States 5 1.3 Classical Beginnings 6

1.4 The Hamiltonian 7

1.5 Born-Oppenheimer Approach 8 1.5.1 Density As Basic Variable 9 1.6 Introduction To DFT 11

1.6.1 The First Hohenberg-Kohn Theorem 11 1.6.2 The Second Hohenberg-Kohn Theorem 13 1.6.3 v-Representability And The Levy Constrained

Search Formalism 13

1.6.4 The Kohn-Sham Equations 14

1.6.5 Approximations For The Exchange-Correlation Functional 16

1.7 Introduction To TD-DFT 17

1.7.1 The Time-Dependent Hohenberg-Kohn-Sham For- malism 18

1.7.2 Adiabatic Approximation 21 1.7.3 Linear-Response Theory 22

1.7.4 Linear-Response TD-DFT And Dyson-Like Equa- tion 25

1.7.5 Casida’s Equation 26 2 state of the art 29

2.1 Comments On My Contribution To This Article 31 2.2 Introduction 33

2.3 Wave-Function Theory 36

2.3.1 Born-Oppenheimer Approximation And Beyond 37 2.3.2 Mixed Quantum/Classical Dynamics 39

2.3.3 Pathway Method 42 2.4 TD-DFT 44

2.5 Perspectives 50 3 wavelets for dft 55

3.1 Comments On My Contribution To This Article 57 3.2 Introduction 59

3.3 Wavelet Theory 63

3.3.1 The Story Of Wavelets 64

xiii

xiv contents

3.3.2 Multiresolution Analysis 66 3.3.3 Wavelets 67

3.3.4 An Example: Simple Haar Wavelets 68 3.3.5 Wavelet Basis 70

3.3.6 The Scaling Basis 70

3.3.7 Interpolating Scaling Functions 71 3.4 Density Functional Theory 74

3.5 Time-Dependent Density Functional Theory 75 3.6 Krylov Space Methods 77

3.7 Numerical Implementation Of DFT In BigDFT 78 3.7.1 Daubechies Wavelets 79

3.7.2 Treatment Of Kinetic Energy 81

3.7.3 Treatment Of Local Potential Energy 82

3.7.4 Treatment Of The Non-Local Pseudopotential 84 3.7.5 The Poisson Operator 85

3.7.6 Numerical Separation Of The Kernel 86 3.8 BigDFT and TD-DFT 87

3.8.1 Calculation Of Coupling Matrix 88 3.9 Results 89

3.9.1 Computational Details 90 3.9.2 Orbital Energies 92 3.9.3 Excitation Energies 94 3.9.4 Oscillator Strengths 96 3.10 Conclusion 98

ii published articles 99 4 spin- flip lr- td -dft 101 4.1 Collinear TD-DFT 101 4.2 Noncollinear TD-DFT 102

4.3 Comments On My Contribution To This Article 102 4.4 Introduction 105

4.5 Spin-Flip TD-DFT 112 4.6 Computational Details 117 4.7 Results 118

4.7.1 C

Ring Opening 119 4.7.2 Photochemical Pathway 125 4.7.3 Conical Intersection 127 4.8 Conclusion 130

5 wavelets for td- dft 135

5.1 Comments On My Contribution To This Article 136 5.2 Introduction 137

5.3 Time-Dependent Density-Functional Theory 140 5.4 Implementation In BigDFT 144

5.5 Validation 148 5.6 Application 155

5.6.1 X-Ray Crystal Structure 155

5.6.2 Spectrum 156 5.7 Conclusion 159 iii work in progress 165

6 molecular orbital symmetry labelling 167 6.0.1 Transition Metal Coordination 168 6.0.2 Crystal Field Theory 171

6.0.3 Octahedral Crystal Field 172

6.0.4 High-Spin (HS) And Low-Spin (LS) 172 6.0.5 Jahn-Teller Distortions 173

6.0.6 Molecular Orbital Theory 174 6.0.7 Ligand Field Theory 175 6.0.8 Spin Crossover 176 6.1 Introduction 179

6.2 Structural Details 180 6.2.1 Basis Sets 180

6.2.2 Computational Details 181 6.3 Results 181

6.3.1 Optimized Geometries And Breathing Coordi- nates 182

6.3.2 Ground State And Excited-State Curves 183 6.4 Conclusion 185

iv concluding remarks 189 7 overall conclusion 191

7.1 Conclusion 191 v appendix 195

a projectors for symmetry blocking 197 a.1 Representation Theory 197

a.2 Symmetry Projection 199 a.3 Symmetry Blocking 200

b molecular orbital symmetry labeling in demon2k 203 b.1 Introduction 203

b.2 Present Implementation 208 b.3 Keywords 210

b.4 Example 210 b.4.1 Input 210 b.4.2 Output 211 b.5 For The Programmer 215

b.5.1 Modified deMon2k(v.2.4.6) Routines 215 b.6 Limitations 216

c lr- tddft calculations in bigdft 217 c.1 Introduction 217

c.2 Running TD-DFT 218

c.2.1 The Input File "input.tddft" 218

c.3 Example 218

xvi contents

c.3.1 Input 218 c.3.2 Output 218

c.3.3 Plotting The Absorption Spectra 219

d analytical derivatives for the excited states us - ing td- dft/tda approach 221

d.1 Analytical Derivatives 221 d.1.1 Preliminaries 222 d.1.2 Turn-Over Rule 223

d.1.3 Analytical Gradients For Ground State 223 d.1.4 Density Matrix Derivatives 225

d.1.5 Solving Linear Response (linear-response (

)) Like Equation 228

d.1.6 Coupled Perturbed Kohn-Sham Equation 230 d.1.7 Analytical Gradients For Excited States Within

TDA 231

d.1.8 Ground State Lagrangian Formalism 232 d.1.9 Z-Vector Method 233

d.1.10 Excited State Lagrangian And Z-Vector Method 234

bibliography 235

Figure 2.1 Schematic representation of potential energy sur- faces for photophysical and photochemical pro- cesses: S

, ground singlet state; S

, lowest ex- cited singlet state; T

, lowest triplet state; absorp- tion (absorption (

)), fluorescence (fluorescence (

)), phosphorescence (phosphorescence (

)), intersystem crossing (intersystem crossing (

)), conical intersection (conical intersection (

)), transition state (transition state (

)). 34 Figure 2.2 Mechanism proposed by Gomer and Noyes in

1950 for the photochemical ring opening of oxi- rane. Reprinted with permission from [302].

Copyright 2008, American Institute of Physics. 47 Figure 2.3 Comparison of Tamm-Dancoff approximation

(

) time-dependent (

)-local density approximation (

) and diffusion Monte Carlo curves for C

ring opening of oxirane. Reprinted with per- mission from [82]. Copyright 2007, American Institute of Physics. 48

Figure 2.4 (a) Cut of potential energy surfaces along reac- tion path of a Landau-Zener (dashed line) and a fewest-switches (solid line) trajectory (black, S

; blue, S

; green, S

; magenta, S

). Both trajecto- ries were started by excitation into the

(n, 3p

) state, with the same geometry and same initial nuclear velocities. The running states of the Landau-Zener and the fewest-switches trajectory are indicated by the red crosses and circles, re- spectively. The geometries of the Landau-Zener trajectory are shown at time a) 0, b) 10, and c) 30 fs. (b) State populations (black, S

; blue, S

; green, S

; magenta, S

) as a function of the fewest-switches trajectory in (a). Reprinted with permission from [302]. Copyright 2008, Ameri- can Institute of Physics. 52

xvii

xviii List of Figures

Figure 2.5 Change of character of the active state along the reactive Landau-Zener trajectory, shown in Fig. 2.4. Snapshots were taken at times (a) 2.6, (b) 7.4, (c) 12.2, and (d) 19.4 fs. For (a) and (b), the running state is characerized by a transition from the highest occupied molecular orbital (

) to the lowest unoccupied molecular orbital (

) plus one (LUMO+1), while for (c) and (d) it is characterized by a HOMO-LUMO transition due to orbital crossing. Note that the HOMO remains the same oxygen nonbonding orbital through- out the simulation. Reprinted with permission from [302]. Copyright 2008, American Institute of Physics. 53

Figure 2.6 Comparison of the S

and S

potential energy surfaces calculated using different methods for the complete active space self-consistent field (

) branching coordinate space. Repro- duced from [177] by permission of the PCCP Owner Societies. 53

Figure 2.7 C

potential energy curves: full calculation (solid lines), two-orbital model (dashed lines).

Reproduced from [177] by permission of the PCCP Owner Societies. 54

Figure 3.1 Schematic overview of the three axes of hierar- chical approximations in the ab initio electronic structure theory. 56

Figure 3.2 Wavelets (bottom) and scaling function (top).

68

Figure 3.3 Haar scaling functions. 68

Figure 3.4 Haar scaling functions and the corresponding wavelets. 69

Figure 3.5 Daubechies scaling function φ(x) and wavelet ϕ(x) of order 16. 81

Figure 3.6 Operations performed in BigDFT 87 Figure 3.7 Example: H

O in a simulation box 91 Figure 3.8 Example: H

O in a simulation box showing fine

grid resolution 91

Figure 3.9 Example: H

O in a simulation box showing coarse grid resolution 91

Figure 3.10 Estimation of integrals within the two-orbital two-electron model (

) model. 96 Figure 3.11 Singlet and triplet excitation energies for CO

calculated using deMon2k 97

Figure 3.12 Singlet and triplet excitation energies for CO

calculated using BigDFT 97

Figure 4.1 Gomer-Noyes mechanism for the ring-opening of oxirane (I). [144] 106

Figure 4.2 Two-orbital model of time-dependent density- functional theory (

) excitations with a triplet reference configuration. 110

Figure 4.3 Dissociation of H

obtained with the present im- plementation of spin-flip (

)-

. The black 1

Σ

curve (circles) is the triplet self-consistent field (

) reference state from which excitations are taken. It is nearly degenerate with the M

= 0 triplet state (not shown) generated by

-

. The red 1

Σ

ground state curve (squares) is a mixture of | σ, σ ¯ | and | σ

, σ ¯

| configurations, with the |σ, σ| ¯ dominating at the equilibrium ge- ometry. The 1

Σ

and 1

Σ

states dissociate to the same neutral "diradical" limit, namely [ H ↑ + H ↓ ↔ H ↓ + H ↑ ]. The blue 2

Σ

state curve (triangles) is also a mixture of |σ, σ| ¯ and |σ

, σ ¯

| configurations, but the "doubly-excited" | σ

, σ ¯

| configuration dominates at the ground state equi- librium geometry. The green curve (diamonds) is the 1

Σ

( σ → σ

) singly-excited state. The 2

Σ

and 1

Σ

(σ → σ

) states dissociate to the same ionic limit, namely [ H

+ H

↔ H

+ H

] . 111

Figure 4.4 Two-orbital model of

excitations with a closed-shell singlet reference configuration. 115 Figure 4.5 Principal frontier molecular orbital spin-flip tran-

sitions involved in the C

ring-opening of oxi- rane beginning from the R

B

[6a

(σ) → 4b

(σ

)]

reference state. 120

Figure 4.6 C

potential energy curves: full calculation (solid lines), two-orbital model (dashed lines).

122

Figure 4.7 Comparison between different methods for the X

A

, 1

B

, and D

A

C

ring-opening poten- tial energy curves:

-

triplet

reference state (black dashed line),

-

(circles), spin- preserving (

)-

(squares), and diffusion Monte Carlo (

) (triangles). All curves have been shifted to give the same ground state en- ergy at a ring-opening angle of 80

. 123 Figure 4.8 Illustration of orbital relaxation effects in

ver-

sus

. 124

xx List of Figures

Figure 4.9 Potential energy curves for asymmetric ring- opening in oxirane calculated with various meth- ods. 127

Figure 4.10 Schematic Walsh diagram showing how the or- bital fillings during asymmetric ring-opening in a normal

calculation. Regions B and C show effective violation of noninteracting v- representability (

). 128

Figure 4.11 Comparison of the S

and S

PESs calculated using different methods for the

branch- ing coordinate space. All but the

-

part of the figure have been adapted from Ref. [302].

See also that reference for a detailed description of the branching coordinates. 129

Figure 4.12 Comparison of the S

→ S

excitation energy surfaces calculated using different methods for the

branching coordinate space. See Ref. [302] for a detailed description of the branch- ing coordinates. 129

Figure 4.13 Schematic Walsh diagram showing how the or- bital fillings vary during asymmetric ring-opening in a

-

calculation. 130

Figure 5.1 N-cyclohexyl-2-(4-methoxyphenyl)imidazo[1,2-α]

pyridin-3-amine (Flugi 6). 140

Figure 5.2 Daubechies scaling function φ ( x ) and wavelet ϕ(x) of order 16. 145

Figure 5.3 Singlet and triplet excitation energies for N

cal- culated using BigDFT 152

Figure 5.4 Comparison of deMon2k and BigDFT N

spec- tra at higher energies. 154

Figure 5.5 Experimental geometry: carbon, orange; nitro- gen, blue; oxygen, red; hydrogen, white. This geometry consists of two nearly planar entities, namely a nearly planar cyclohexane (C

H

-) ring and the rest of the molecule which rests in a plane perpendicular to the plane of the cyclo- hexane 156

Figure 5.6 Comparison of theoretical and measured absorp- tion spectra for Flugi 6 (left y-axis). The mag- nitude of the BigDFT curve has been divided by a factor of ten (see text). Both theoretical and experimental curves show qualitative agree- ment with the oscillator strength stick-spectra which however is in different units (right y-axis).

157

Figure 6.1 Linear [D

](Image found on the web [8]) 168

Figure 6.2 Trigonal planar [D

] (Image found on the web [8]) 168

Figure 6.3 Trigonal pyramidal [C

] (Image found on the web [8]) 169

Figure 6.4 Square planar [D

] ( Image found on the web [8]) 169

Figure 6.5 Tetrahedral [T

] ( Image found on the web [8]) 169

Figure 6.6 Trigonal bipyramidal [D

] ( Image found on the web [8]) 170

Figure 6.7 Square pyramidal [C

] ( Image found on the web [8]) 170

Figure 6.8 Octahedral [D

] ( Image found on the web [8]) 170

Figure 6.9 The five atomic d-orbitals on the Cartesian axis.

There are two atomic d-orbitals that point direct along the Cartesian axis: d

(which points along the z-axis) and d

(which has lobes on both x- and y-axes.) The other three atomic d-orbitals (d

, d

and d

), have lobes in between the Cartesian axis. A 45 ◦ rotation of d

along the z-axis results in d

, and a 90 ◦ rotation of d

along the z-axis results in d

. Image found on the web [7] 171

Figure 6.10 Illustration of a LS and HS d

system in an octa- hedral environment 172

Figure 6.11 Illustration of alternative spin states: LS, inter- mediate spin and HS of a d

system in an octa- hedral environment 173

Figure 6.12 Jahn-Teller distortions along the z-axis. The Jahn-Teller theorem states that any non-linear molecule system in a degenerate electronic state will be unstable and will undergo distortion to form a system of low symmetry and low energy, thereby removing the degeneracy 174

Figure 6.13 Schematic representation of the two possible spin states for iron(II) and iron(III) coordination compounds in an octahedral environment 176 Figure 6.14 Starting geometry for optimizations for S=5/2,3/2

and 1/2 spin states of [Fe(H

O)

]

180 Figure 6.15 Elementary pairing energy argument 182 Figure 6.16 Ground-state curves 184

Figure 6.17 Singlet Walsh diagram 185

Figure 6.18 Triplet Walsh diagram 186

Figure 6.19 Quintet Walsh diagram 186

Figure B.1 Example of symmetry: Tamil mandala painted on a roof inside a temple located in Mauritius Island. (Found on the web.) 203

Figure B.2 Brief schematic of the history of the deMon suite of programs. Taken from the web site [10] 209 Figure B.3 Molecular orbitals of H

O 214

Figure C.1 Absorbtion spectrum of Nitrogen 219

L I S T O F TA B L E S

Table 3.1 Basis set dependence of the

and

energies and of the

-

gap (eV) calcu- lated using deMon2k. 93

Table 3.2 Basis set dependence of the

and

energies and of the

-

gap (eV) calcu- lated using BigDFT. 94

Table 3.3 Comparison of lowest excitation energies of CO (in eV) calculated using BigDFT and deMon2k and with experiment. 95

Table 3.4 Estimations of integrals (in eV) within the

. 96

Table 3.5 Comparison of experimental A

Π energies (eV) and oscillator strengths with

-

/

ex- perimental A

Π energies (eV) and degeneracy- weighted oscillator strengths (unitless.) 98 Table 5.1 Basis set dependence of the

and

energies and of the

-

gap (eV) calcu- lated using deMon2k. 150

Table 5.2 Basis set dependence of the

and

energies and of the

-

gap (eV) calcu- lated using BigDFT. 151

Table 5.3 Comparison of the nine lowest excitation en- ergies of N

(in eV) calculated using different programs and with experiment. 153

Table 5.4 Experimental geomentry (Cartesian coordinates in Å) for the Flugi 6 162

Table 5.5 density-functional theory (

) optimized ge- ometries (Cartesian coordinates in Å) of Flugi 6. Calculations performed at the

level of theory. 163

Table 5.6 Singlet excitation energies ( hω

, in eV) up to

−ǫ

= 4.8713 eV, oscillator strength (f

, unit- less) and assignment. 164

xxii

Table 6.1 Average Fe-O bond lengths of the three spin states. 182

Table 6.2 Energies of High, Intermediate, and Low Spin State Complexes at Optimized Geometries . 183 Table B.1 C

group character table. (See for example

Ref. [83].) 210

A C R O N Y M S

For the readers convenience we give a list of the abbreviations used in this thesis in alphabetical order:

AA

adiabatic approximation

ABS

absorption

ADF

Amsterdam density-functional

ALDA

adiabatic LDA

AO

atomic orbitals

APW

augmented plane wave

a.u

atomic units

AX

avoided crossing

BO

Born-Oppenheimer

CASSCF

complete active space self-consistent field

CC

coupled cluster

CDFT-CI

constrained density-functional theory-CI

CFSE

crystal field stabilization energy

CFT

crysal field theory

CI

configuration interaction

CIS

configuration interaction singles

crmult

coarse grid multiplier

CX

conical intersection

DC

derivative nonadiabatic coupling vector

DFT

density-functional theory

DIIS

discret inversion in the iterative subspace

DM

density matrix

DMC

diffusion Monte Carlo

DMSO

dimethyl sulphur dioxide

DZVP

double-zeta-valance polarization

xxiv

EXX

exact exchange

FD

finite difference

FE

finite element

FLUO

fluorescence

frmult

fine grid multiplier

FWHM

full-width at half-maximum

GGA

generalized gradient approximation

GTH

Goedecker-Teter-Hutter

GTOs

Gaussian-type orbitals

Ha

Hartree energy

HF

Hartree-Fock

HGH

Hartwigsen-Goedecker-Hutter

HK

Hohenberg-Kohn

HOMO

highest occupied molecular orbital

HS

high-spin

ISF

interpolating scaling function

ISX

intersystem crossing

KS

Kohn-Sham

LCAO

linear-combination of atomic orbital

LDA

local density approximation

LFT

ligand field theory

LGOs

ligand group orbitals

LMTO

linear muffin-tin orbital

LS

low-spin

LUMO

lowest unoccupied molecular orbital

MBPT

many-body perturbation theory

MCSCF

multi-configurational self-consistent field

MO

molecular orbitals

MP

Møller-Plesset

xxvi acronyms

MRA

multiresolution analysis

MRCI

multireference configuration interaction

NVR

noninteracting v-representability

OEP

optimized effective potential

PES

potential energy surfaces

PHOS

phosphorescence

QMC

quantum Monte Carlo

RI

resolution-of-identity

ROKS

restricted open-shell Kohn-Sham

RPA

random-phase approximation

S

singlet

SCF

self-consistent field

SCO

spin crossover

SF

spin-flip

SI

international System

SIC

self-interaction correction

SOS

sum-over-states

SP

spin-preserving

T

triplet

TD

time-dependent

TDA

Tamm-Dancoff approximation

TD-DFT

time-dependent density-functional theory

TOTEM

two-orbital two-electron model

LR

linear-response

TRK

Thomas-Reiche-Kuhn

TS

transition state

TZVP

triple-zeta-valance polarization

UGD

unscaled gradient difference vector

xc

exchange-correlation

T H E O R E T I C A L M O T I VAT I O N

B A C K G R O U N D M AT E R I A L 1

Many molecular-quantum-mechanics research seminars, theses, and reviews have began by quoting a remark made by Paul Dirac:

"The underlying physical laws necessary for the mathe- matical theory of a larger part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equa- tions much too complicated to be soluble." [Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 123, No.

792 (6 April 1929)]

While this remark is as true now as it was then (we are still unable to afford exact solutions to the equations), the development of super- computers has enabled increasingly accurate solutions to be obtained.

Presented in this chapter are: (i) the equations of the molecular quan- tum mechanics which explain the majority of chemistry referred to by Dirac, (ii) the basis upon which we approximate their solution, and (iii) the ways in which we seek to improve such approximations.

In the context of this thesis, emphasis is placed on

, and

treatments are introduced for completeness and for later discussion.

1.1 synopsis

The spirit of this thesis is very much that attributed to Dirac. The goal is to find more efficient accurate ways to solve the Schrödinger equation in order to extract chemically-useful information. Part of the answer involves improving the numerical methods used in present- day electronic structure calculations and part of the answer involves seeking practical approximations. The purpose of this synopsis is to explain–in broad strokes–the specific approach to be taken during my thesis work and the general organization of my thesis.

The approach taken in my work is to use

for ground states and

LR TD-DFT

for electronic excited states. The emphasis on

, rather than wave-function-based approaches, is a question of efficiency.

calculations simply scale better with system size than do wave-function methods of comparable accuracy.

-

is now a well-established

1 The quote is typically given out of context as presented here. The rest of the text actually goes on to speak about the need for approximations: "It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation."

3

4 background material

way to extract information about electronic excited states. Nevertheless much contemporary effort is currently being devoted to improving

DFT

and

, both from the point of view of commonly-used underlying approximations and from the point of view of improved numerical algorithms. The focus of this thesis is on

rather than conventional ground-state

. Algorithms and approximations are proposed, tested, and (in some cases) applied to previously unstudied systems. High-lights include:

• Testing an implementation of

-

for the computation of potential energy surfaces (

) with comparison against mul- tireference wave-function methods at an accuracy comparable to that of multireference methods.

• Implementation and exploration of wavelet-based

-

calculations.

The thesis is "by articles," meaning that it includes entire published articles or articles submitted for publication. The reason for doing so is that some of the work was only possible because of the contributions of several co-authors and this seemed the best way to acknowledge them. Each article is prefaced with an introduction placing the work in the context of this thesis and describing my contribution to that article. Of course, this type of thesis has the drawback that many of the articles retain redundant introductory material. Notation may also sometimes differs from article to article. That is inevitable when choosing this type of format and the readers are thanked ahead of time for their understanding and indulgence.

The thesis is mainly divided into four parts. Part I (Chapters 1– 3) consists of background material. Chapter 1 consists of an "elementary"

review of "textbook material" needed to keep the thesis self-contained and accessible to (say) a new doctoral student. Chapters 2 and 3 review, respectively, the state of the art for

investigations of photochemistry and wavelet algorithms for solving the Kohn-Sham (

) equations of

.

Part II (Chapters 4 and 5) are original scientific contributions which are either already published articles or have been submitted for pub- lication. The performance of the deMon2k implementation of

-

-

TD-DFT

is investigated in Chapter 4 for its ability to describe key

features needed for a realistic description of photochemical reactions.

Chapter 5 describes our implementation, testing, and application of

LR

-

in the wavelet code BigDFT. Implementation details are given in appendix C following the theoretical details in appendix A.

Part III (Chapters 6 presents on-going research. Chapter 6 de-

scribes my personal experience implementing the molecular-orbital

symmetry labeling in the Grenoble version of deMon2k whose docu-

mentation is described in appendix B and its successful application to

[Fe(H

O)

]

–a common example in inorganic chemistry books dis- cussing ligand field theory but also a molecule with rare T

symmetry.

Part IV (Chapter 7) concludes and points out how our work might best be continued in the future.

Appendix D presents the equations that we have worked out for implementing analytic gradients for

excited states in deMon2k and in BigDFT.

1.2 atomic units

We use a system of Hartree energy (

) atomic units (

) throughout this dissertation unless otherwise denoted. Atomic units are based upon Gaussian rather than international System (

) units, so that factors of 4πǫ

never appear. There are also Rydberg atomic units but they rarely work anymore. In atomic units,

e = m

= h = 1

where e is the elementary charge on the proton, m

is the electron mass, and h Planck’s constant divided by 2π. Conversion to other units can be obtained by dimensional analysis. For example, distance is in units of h

/m

e

and called the Bohr radius a

, and energy is in units of e

/a

and is called

(E

). The equations expressed in

units Coulomb’s constant 1/4πǫ

also have magnitude 1.

1.2.1 Naming Excited States

There are various notations for the electronic states of a molecule and the transitions between them. The most widely used nomenclatures are the enumerative one and Kasha’s.

The enumerative nomenclature is based on the energetic order of the states and their multiplicities. The electronic state with the lowest energy defines the ground state, and the adiabatic energies of the other states determine the corresponding labels. Thus, in this notation the singlet ground state is denoted by S

, whereas the excited states are expressed by the successive numbers S

, S

, S

,...,S

. A similar formula is used for the triplet states, which are denoted by T

, T

,...,T

. The excitations are expressed by S

→ S

, where the arrow indicates the direction of excitation. It is worthwhile to mention that, since the nature of the states can change along the

, the enumerative nomenclature can sometimes lead to confusion.

Kasha’s nomenclature only specifies the nature of the state involved in the transition. The symbols π,σ, and n characters are used to denote unsaturated, sigma and non-bonding occupied orbitals, respectively.

The same symbols with an added " ∗ " (π

, σ

) are used to refer to

the corresponding antibonding orbitals. As in the former case, the

excitations are expressed as π → π

. With this nomenclature the states

6 background material

present smooth energetic profiles along the

. We will use both nomenclatures indistinctively throughout this thesis. For the sake of shortness, excitations will either be denoted as π, π

or π − π

. Basically the enumerative nomenclature labels adiabatic PESs while Kasha’s notation labels diabatic

.

1.3 classical beginnings

The classical mechanics which is so strongly associated with the name of Isaac Newton grew out of our everyday experience with balls, pendulums, boats on water–macroscopic objects of all kinds– and the realization that these laws also describe the motion of heavenly bodies. However classical mechanics fails miserably when describing objects as small as atoms and molecules. In 1911, Rutherford [281]

proposed that an atom must contain a small massive center called the nucleus. He believed the nucleus contained all the positive charge of the atom and was surrounded by orbiting electrons with equal and opposite charge which moved around the nucleus much like planets around the sun. Later, Bohr [47] suggested that electrons could not spiral inwards out of these orbits by emitting continuous radiation as they were only allowed to emit quanta. This Rutherford-Bohr model mixes the classical physics idea of electrons as particles in orbits with concepts of energy quanta, and while this model successfully explains the emission spectrum of hydrogen atom, it is incorrect. Nevertheless, it was the introduction of quantum theory to chemistry, and this was where quantum mechanics began to take over.

Quantum mechanics offers chemists the possibility of simulating and hence understanding molecular structure, spectroscopy, and reac- tivity. However, only the two particle hydrogen atom (H) has a closed solution and only a few other systems may be considered to have essentially numerically exact solutions (e.g., H

and He). The rest of this chapter tries to give an elementary introduction to the basic quantum mechanics used in this thesis and to practical approximation needed for treating nontrivial problems.

Quantum mechanics was born when two important theories come together to give a clear explanation of the hydrogen spectrum. Louis de Broglie postulated that every particle had a wavelength associated with its momentum p,

λ = h p

As nonemitting charged entities, electrons are restricted to orbits with integral wavelengths,

2πr = nλ so their angular momentum is quantized

2πr = nh

p ⇒ pr = n h

By combining these relations with the equation balancing centrifugal and Coulombic forces,

p

mr = e

(4πǫ

)r

we have two equations in two unknowns which may be solved to give p = me

( 4πǫ

) h h r = (4πǫ

)n

h

me

, (1.1)

since

E = p

2m − e

(4πǫ

)r , then

E = − m

e

2(4πǫ

)

n

h

= − m

e

8ǫ

n

h

. In Gaussian units this is

E = − m

e

2n

h

and in atomic units

E = − 1 2n

.

The energy gaps between these stable orbitals compose the hydrogen spectrum. Beyond the hydrogen atom, quantum mechanics cannot be solved exactly but numerical solutions of H

and He are essentially exact.

Using the de Broglie wavelength, Erwin Schrödinger substituted the momentum of an electron for the frequency term in the classical wave equation and derived his famous differential equation. For the hydrogen atom, the Schrödinger equation reads (in

)

− 1

2 ∇

− 1 r

ψ(r) = ǫψ(r) , (1.2)

The Schrödinger equation defines kinetic h ψ| −

∇

|ψ i and potential h ψ | −

| ψ i energies in terms of electronic wave functions, ψ ( r ) . P. A.

M. Dirac showed that Werner Heisenberg’s initial matrix mechanics form of quantum mechanics is in fact just an alternative linear algebra form of Schrödinger’s later wave mechanics formulation of quantum mechanics.

1.4 the hamiltonian

First of all, I am going to describe the Hamiltonian of an N-electron

M-nuclei system. In principle the Hamiltonian contains all the physics

8 background material

of this many-body system. By solving the Schrödinger equation with this Hamiltonian, we should then be able to derive all the observable quantities. Its form is,

H ˆ = X

i

− 1 2 ∇

+ X

α

− 1 2M

∇

+ 1

2 X

i6=j

1 r

+ 1 2

X

α6=β

Z

Z

R

− X

i,α

Z

|r

− R

| . (1.3) The first term is the electronic kinetic energy operator, where ∇

is the Laplacian acting over the electronic coordinates {r

} with an electronic mass m. The second term is the operator corresponding to the kinetic energy of the nuclei, where ∇

is the Laplacian acting over the nuclear coordinates {R

} with the mass of nuclei M

. The third and fourth terms are the pairwise electrostatic electron-electron and nucleus-nucleus interactions respectively, where, r

= |r

− r

| and R

= |R

− R

| are the electron-electron and nucleus-nucleus separations of the pairs which are being considered, and Z

represents the charge of the αth nucleus. Finally, the fifth term corresponds to the electron-nuclear attraction.

From the Hamiltonian given above, is clear that the number of independent variables in the corresponding Schrödinger’s equation is determined by the number of particles involved. Therefore, an exact solution to such kind of equation is not possible for realistic systems.

Hence, when dealing with this kind of problems, people very often try to work them out by applying different successive approximations in order to model the physics of the system, which, in some cases, can compromise the accuracy of the final result or at least provide results that are not general. In some cases, this leads to empirical models (i.e., models containing external parameters) which will work well for only a few kinds of systems and external conditions. (Note, however, that empirical methods are largely avoided in this thesis, which favours a first-principles density-functional theory approach.)

Very frequently, the first approximation that people do is the adi- abatic (or Born-Oppenheimer (

) [48]) approximation. This one is usually not very critical in terms of loss of accuracy, and instead simplifies considerably our problem so, next, I am going to present a short description of what it is about.

1.5 born- oppenheimer approach

If we divide the system into light particles (electrons) and heavy ones

(atomic nuclei) and think classically, in thermodynamic equilibrium

the mean value for the kinetic energy of both kind of particles is of

the same order [49, 50] but, due to the large mass difference between

nucleons (i.e., protons+neutrons) and electrons, the electronic speed

very much exceeds nuclear speeds ( by approximately two orders of

magnitude.) Then, for every modification in the position of the atomic nuclei an almost instantaneous rearrangement of the electrons occurs, following the new nuclear positions. This allows us to consider, at least to a first approximation, the movement of the electrons as if they were in field of fixed nuclei. While studying the movement of the nuclei, on the contrary only the potential originating from the mean electronic spatial distribution (not an instantaneous one) must be taken into account. When this physical approximation is formulated in quantum mechanics, it is known as the adiabatic or

approximation.

In the

approximation, the nuclear Schrödinger equation is written as,

− X

α

1

2M

∇

+ V

(R)

!

Φ(R) = ǫΦ(R) , (1.4) where R = { R

} is the set of all the nuclear coordinates and V

( R ) the clamped-ion energy of the system, which is often referred to as the Born-Oppenheimer potential energy surface, and ǫ are the atomic eigenvalues. In practice, V

( R ) = E

( R ) + V

( R ) is the ground-state energy of a system of interacting electrons moving in the field of fixed nuclei, which obeys the Schrödinger equation

H

(r; R)φ

(x; R) = E

(R)φ

(x; R) . (1.5) where the Hamiltonian–which acts on to the electronic variables and depends only parametrically on R–reads

H

(r; R) = − 1 2

X

i

∇

+ 1 2

X

i6=j

1 r

− X

i,α

Z

|r

− R

| , (1.6) plus the nuclear-nuclear repulsion energy,

V

(R) = X

α<β

Z

Z

|R

− R

| . (1.7)

This could be mistaken as a simple rearrangement of equation (1.3), but it is important to notice that now the electronic part is decoupled from the rest and can be solved independently, using the set of nuclear positions R, only as parameters. Explicit inclusion of the coupling of the electronic and nuclear degress of freedom beyond the Born- Oppenheimer approximation and present in the full Schrödinger equation is important for simulations of some types of photoprocess.

1.5.1 Density As Basic Variable Within the

approximation,

V

(R) = E

(R) + V

(R) . (1.8)

10 background material

while the nuclear-nuclear repulsion V

( R ) is a relatively simple ob- ject, the electronic energy E

(R) for nuclei frozen in the configuration R is still very complicated. A priori E

(R) depends upon φ(x; R) where x = { x

} and x

= ( r

, σ

) are the space plus spin coordinates of N elec- trons. Nevertheless chemists have long analyzed chemical reactivity in terms of atomic charges–which is to say, in terms of the charge density

ρ(r

; R) = X

σ

Z

|φ

(x

; R)|

dx

dx

...dx

. (1.9) Why should this work? Part of the answer is that the electron-nuclear attraction depends only upon the charge density,

V

( R ) = − Z

ρ ( r; R ) X

α

Z

|r − R

|

!

d ~ r , (1.10)

as does the classical electron-electron repulsion energy V

( R ) = 1

2

Z ρ(r

; R)ρ(r

; R)

| r

− r

| dr

dr

. (1.11) For atoms, we can also obtain the electronic kinetic energy from the virial theorem [268] as

T

(R) = h φ

(R)| − 1 2

X ∇

|φ

(R) i

= − 1

2 V

(R) . (1.12)

Indeed T

(R) + V

(R) + V

(R) accounts for most of E

(R) but it it still not an accurate-enough approximation for most problems in chemical and solid-state physics.

A solution in principle was given by E. Bright Wilson [36], who pointed out that (i) integration over ρ gives the number of electrons, N, (ii) the cusps in ρ give the nuclear positions, R, and (iii) the derivatives of ρ at the cusps give the nuclear charges, {Z

} . Hence the ground- state density of a molecule contains all of the information necessary to reconstruct the electronic Hamiltonian H

(r; R) and hence to determine all of the electronic properties of the system by "just" solving the electronic Schrödinger equation. In particular, the ground-state energy is a functional of the ground-state charge density

E

( R ) = E

( R )[ ρ

( ~ r; R )] . (1.13) In 1964, Hohenberg and Kohn provided a more general mathematical proof of this same result, so beautifully motivated by E. Bright Wilson.

The Hohenberg-Kohn (

) theorems and the Kohn–Sham formulation

of density–functional theory (DFT) are the subject of the next section.

1.6 introduction to dft

Over the last 45 years

has become one of the standard methods for calculations in several branches of physics and chemistry. Among all the other approaches to electronic structure calculations, such as configuration interaction (

), coupled cluster (

) and Møller-Plesset (

) perturbation theory, the rather special place of

becomes directly clear from the fundamentals, as first formulated in 1964 by Hohenberg and Kohn [172].

1.6.1 The First Hohenberg-Kohn Theorem

Every observable quantity of a stationary quantum me- chanical system is determined by the ground-state density alone.

In other words, the aim of

is not to obtain a good approximation to the ground-state wave function of the system, but rather to find the energy of the system as a functional of the density, without any reference to the wave function. This proof, that all observables of a many electron system are unique functionals of the electron density, provides the theoretical basis for

.

Consider a nonrelativistic N-electron system in the Born-Oppenheimer approximation. The Hamiltonian ˆ H in the Schrödinger equation

H ˆ

Ψ ( x

, x

, ....x

) = EΨ ( x

, x

, ...., x

) , (1.14) consists of the kinetic energy ˆ T, the nuclear-electron interaction ˆ v

, and the electron-electron interaction ˆ v

.

In the

theorem the one-to-one mapping between the electron density ρ

ρ(r

) = X

σ₁=↑↓

N Z

|Ψ(x

, x

, ..., x

)|

dx

..., dx

, (1.15) and the external potential ˆ v = P

i

v(r

) (which is typically just the nuclear attraction v(r

) = − P

i

P

αZ_α

r_iα

or could be more general) is proved. The mapping

ˆ

v −−−−−−−→

Ψ −−−−−−−→

ρ , (1.16) is rather simple and straight forward. Each ˆ v connects to a wave func- tion, Ψ, by solving Schrödinger equation (1.14), and the corresponding density ρ can be found by integrating the square of the wavefunction Eq. (1.15).

The proof of the mapping in other direction (that ρ determines ˆ v) ˆ

v ← −

Ψ ←

− ρ , (1.17)

12 background material

is done in two steps. In each case, the proof is by contradiction: (a) If ˆ v and ˆ v

differ by more than a constant C, they will not lead to the same wavefuction Ψ.

(b) If the ground state Ψ of ˆ H and Ψ

of ˆ H

are different, they cannot lead to the same density ρ.

Let us first prove the lemma (a). For a nondegenerate ground state it follows that,

" T ˆ + v ˆ

+ v ˆ

|Ψ i = E

|Ψ i , (1.18)

" T ˆ + v ˆ

+ v ˆ

|Ψ

i = E

|Ψ

i , (1.19) Now we assume that Ψ = Ψ

, and one gets

( v ˆ − v ˆ

)| Ψ i = ( E

− E

)| Ψ i , (1.20) which immediately leads to ˆ v = v ˆ

+ C at least where Ψ 6 = 0, in contradiction with the assumption made in ( a ) .

We now prove the lemma (b). Using the variational principle, in the case for (b) one proves that if Ψ 6 = Ψ

. This implies that ρ(r) 6 = ρ

(r) . Without loss of generality we may choose Ψ and Ψ

such that

E

= h Ψ | H ˆ | Ψ i < h Ψ

| H ˆ | Ψ

i ,

= h Ψ

| H ˆ

+ v ˆ − v ˆ

|Ψ

i ,

= E

+ Z

ρ

( r )[ v ( r ) − v

( r )] dr . (1.21) and similarly

E

= h Ψ

| H ˆ

|Ψ

i < h Ψ| H ˆ

|Ψ i ,

= h Ψ| H ˆ + v ˆ

− v|Ψ ˆ i ,

= E

+ Z

ρ(r)[v

(r) − v(r)]dr . (1.22) Assuming that ρ(r) 6 = ρ

(r), the combination of the equations (1.21) and (1.22) leads to the following contradiction

E

+ E

< E

+ E

. (1.23) Therefore the maps between ˆ v, Ψ and ρ are bijective (i.e., one-to-one).

ˆ

v ←→ Ψ ←→ ρ , (1.24)

and as a consequence of the bijective map Ψ ←→ ρ, every observable O of the system is a unique functional of the density

h Ψ[ρ]| O|Ψ[ρ] ˆ i = O[ρ] . (1.25)